Stochastic Evolution Equations Research Articles

Stationary distributions of second order stochastic evolution equations with memory in Hilbert spaces

In this paper, we consider stationarity of a class of second-order stochastic evolution equations with memory, driven by Wiener processes or Lévy jump processes, in Hilbert spaces. The strategy is to formulate by reduction some first-order systems in connection with the stochastic equations under investigation. We develop asymptotic behavior of dissipative second-order equations and then apply them to time delay systems through Gearhart–Prüss–Greiner’s theorem. The stationary distribution of the system under consideration is the projection on the first coordinate of the corresponding stationary results of a lift-up stochastic system without delay on some product Hilbert space. Last, two examples of stochastic damped wave equations with memory are presented to illustrate our theory.

A randomized and fully discrete Galerkin finite element method for semilinear stochastic evolution equations

In this paper the numerical solution of non-autonomous semilinear stochastic evolution equations driven by an additive Wiener noise is investigated. We introduce a novel fully discrete numerical approximation that combines a standard Galerkin finite element method with a randomized Runge-Kutta scheme. Convergence of the method to the mild solution is proven with respect to the $L^p$-norm, $p \in [2,\infty)$. We obtain the same temporal order of convergence as for Milstein-Galerkin finite element methods but without imposing any differentiability condition on the nonlinearity. The results are extended to also incorporate a spectral approximation of the driving Wiener process. An application to a stochastic partial differential equation is discussed and illustrated through a numerical experiment.

On Jumps Stochastic Evolution Equations With Application of Homogenization and Large Deviations

We consider a class of jumps and diffusion stochastic differential equations which are perturbed by to two parameters:  ε (viscosity parameter) and δ (homogenization parameter) both tending to zero. We analyse the problem taking into account the combinatorial effects of the two parameters  ε and δ . We prove a Large Deviations Principle estimate for jumps stochastic evolution equation in case that homogenization dominates.

Optimal regularity of stochastic evolution equations in M-type 2 Banach spaces

In this paper, we prove the well-posedness and optimal trajectory regularity for the solution of stochastic evolution equations driven by general multiplicative noises in martingale type 2 Banach spaces. The main idea of our method is to combine the approach in [9] dealing with Hilbert setting and a version of Burkholder inequality in M-type 2 Banach space. Applying our main results to the stochastic heat equation gives a positive answer to an open problem proposed in [10].

Existence and stability of μ-pseudo almost automorphic solutions for stochastic evolution equations

We introduce a new concept of μ-pseudo almost automorphic processes in p-th mean sense by employing the measure theory, and present some results on the functional space of such processes like completeness and composition theorems. Under some conditions, we establish the existence, uniqueness, and the global exponentially stability of μ-pseudo almost automorphic mild solutions for a class of nonlinear stochastic evolution equations driven by Brownian motion in a separable Hilbert space.

Well-posedness and dynamics of impulsive fractional stochastic evolution equations with unbounded delay

This paper is concerned with the well-posedness and dynamics of delay impulsive fractional stochastic evolution equations with time fractional differential operator α ∈ (0, 1). After establishing the well-posedness of the problem, and a result ensuring the existence and uniqueness of mild solutions globally defined in future, the existence of a minimal global attracting set is investigated in the mean-square topology, under general assumptions not ensuing the uniqueness of solutions. Furthermore, in the case of uniqueness, it is possible to provide more information about the geometrical structure of such global attracting set. In particular, it is proved that the minimal compact globally attracting set for the solutions of the problem becomes a singleton. It is remarkable that the attraction property is proved in the usual forward sense, unlike the pullback concept used in the context of random dynamical systems, but the main point is that the model under study has not been proved to generate a random dynamical system.

Central limit theorems and minimum-contrast estimators for linear stochastic evolution equations

ABSTRACTCentral limit theorems and asymptotic properties of the minimum-contrast estimators of the drift parameter in linear stochastic evolution equations driven by fractional Brownian motion are studied. Both singular ( and regular ( types of fractional Brownian motion are considered. Strong consistency is achieved by ergodicity of the stationary solution. The fundamental tool for the limit theorems and asymptotic normality (shown for Hurst parameter ) is the so-called 4th moment theorem considered on the second Wiener chaos. This technique provides also the Berry-Esseen-type bounds for the speed of the convergence. The general results are illustrated for parabolic equations with distributed and pointwise fractional noises.

Optimal control of a class of semi‐linear stochastic evolution equations with applications

In this study, the authors derive a new type of maximum principle for both stochastic evolution and parabolic type stochastic partial differential systems. The systems only require the Hamiltonian functional to be concave in the state variable rather than in both state and control variables. They also show a connection between these two types of systems. Finally, examples are given to illustrate the authors' theoretical results.

Well-posedness of stochastic Riccati equations and closed-loop solvability for stochastic linear quadratic optimal control problems

We study the closed-loop solvability of a stochastic linear quadratic optimal control problem for systems governed by stochastic evolution equations. This solvability is established by means of solvability of the corresponding Riccati equation, which is implied by the uniform convexity of the quadratic cost functional. At last, conditions ensuring the uniform convexity of the cost functional are discussed.

Optimal relaxed control of stochastic hereditary evolution equations with Lévy noise

Existence theory of optimal relaxed control problem for a class of stochastic hereditary evolution equations driven by Lévy noise has been studied. We formulate the problem in the martingale sense of Stroock and Varadhan to establish existence of optimal controls. The construction of the solution is based on the classical Faedo–Galerkin approximation, the compactness method and the Jakubowski version of the Skorokhod theorem for nonmetric spaces, and certain compactness properties of the class of Young measures on Suslin metrizable control sets. As application of the abstract theory, Oldroyd and Jeffreys fluids have been studied and existence of optimal relaxed control is established. Existence and uniqueness of a strong solution and uniqueness in law for the two-dimensional Oldroyd and Jeffreys fluids are also shown.

Approximation of the interface condition for stochastic Stefan-type problems

We consider approximations of the Stefan-type condition by imbalances of volume closely around the inner interface and study convergence of the solutions of the corresponding semilinear stochastic moving boundary problems. After a coordinate transformation, the problems can be reformulated as stochastic evolution equations on fractional power domains of linear operators. Here, the coefficients might fail to have linear growths and might be Lipschitz continuous only on bounded sets. We show continuity properties of the mild solution map in the coefficients and initial data, also incorporating the possibility of explosion of the solutions.

The Kushner-Stratonovich stochastic method for a master robot of a tele-robotic system: beyond extended Kalman filtering

The surgical robotic system, a tele-robotic system, comprises the operator, master-slave configuration in combination with the object. For the estimation and control of the surgical robotic system, the master-slave set up is the subject of greater robotic research activities as well as has proven promising. In this paper, the multi-dimensional stochastic differential equation as well as the higher-order nonlinear stochastic filter beyond extended Kalman filtering for a master robot of the surgical robotic system are developed. The higher-order nonlinear filtering preserves the greater qualitative characteristics of nonlinear stochastic systems by accounting observation correction terms in stochastic evolution equations. First, the Lagrangian mechanics is exploited to develop the vector stochastic model of the master robot. Secondly, this paper utilises the Kushner-Stratonovich equation, for filtering of the master robot noise equations. This paper demonstrates comparison between the higher-order Kushner-Stratonovich filter, extended Kalman filter and true state trajectories as well.

The Kushner-Stratonovich stochastic method for a master robot of a tele-robotic system: beyond extended Kalman filtering

The surgical robotic system, a tele-robotic system, comprises the operator, master-slave configuration in combination with the object. For the estimation and control of the surgical robotic system, the master-slave set up is the subject of greater robotic research activities as well as has proven promising. In this paper, the multi-dimensional stochastic differential equation as well as the higher-order nonlinear stochastic filter beyond extended Kalman filtering for a master robot of the surgical robotic system are developed. The higher-order nonlinear filtering preserves the greater qualitative characteristics of nonlinear stochastic systems by accounting observation correction terms in stochastic evolution equations. First, the Lagrangian mechanics is exploited to develop the vector stochastic model of the master robot. Secondly, this paper utilises the Kushner-Stratonovich equation, for filtering of the master robot noise equations. This paper demonstrates comparison between the higher-order Kushner-Stratonovich filter, extended Kalman filter and true state trajectories as well.

NEWTON S METHOD FOR STOCHASTIC FUNCTIONAL EVOLUTION EQUATIONS IN HILBERT SPACES

We apply Newton’s method to stochastic functional evolution equations in Hilbert spaces using semigroup methods. The first-order convergence is based on our generalization of the Gronwall-type inequality. We also establish a second-order convergence in a probabilistic sense.

Partially Observed Stochastic Evolution Equations on Banach Spaces and Their Optimal Lipschitz Feedback Control Law

In this paper we consider optimal feedback control problems for a general class of nonlinear partially observed stochastic evolution equations on Banach spaces. The system is governed by a pair of ...

Erratum: Stochastic Evolution Equations for Large Portfolios of Stochastic Volatility Models

In the article "Stochastic evolution equations for large portfolios of Stochastic Volatility models" (Arxiv:1701.05640) there is a mistake in the proof of Theorem 3.1. In this erratum we establish a weaker version of this Theorem and then we redevelop the regularity theory for our problem accordingly. This means that most of our regularity results are replaced by slightly weaker ones. We also clarify a point in the proof of a correct result.

Malliavin regularity and weak approximation of semilinear SPDEs with Lévy noise

We investigate the weak order of convergence for space-time discrete approximations of semilinear parabolic stochastic evolution equations driven by additive square-integrable Levy noise. To this end, the Malliavin regularity of the solution is analyzed and recent results on refined Malliavin-Sobolev spaces from the Gaussian setting are extended to a Poissonian setting. For a class of path-dependent test functions, we obtain that the weak rate of convergence is twice the strong rate.

Weak convergence rates for stochastic evolution equations and applications to nonlinear stochastic wave, HJMM, stochastic Schrödinger and linearized stochastic Korteweg–de Vries equations

We establish weak convergence rates for noise discretizations of a wide class of stochastic evolution equations with non-regularizing semigroups and additive or multiplicative noise. This class covers the nonlinear stochastic wave, HJMM, stochastic Schr\"odinger and linearized stochastic Korteweg-de Vries equation. For several important equations, including the stochastic wave equation, previous methods give only suboptimal rates, whereas our rates are essentially sharp.

Martingale solutions of nematic liquid crystals driven by pure jump noise in the Marcus canonical form

In this work we consider a stochastic evolution equation which describes the system governing the nematic liquid crystals driven by a pure jump noise in the Marcus canonical form. The existence of a martingale solution is proved for both 2D and 3D cases. The construction of the solution relies on a modified Faedo–Galerkin method based on the Littlewood–Paley-decomposition, compactness method and the Jakubowski version of the Skorokhod representation theorem for non-metric spaces. We prove that in the 2-D case the martingale solution is pathwise unique and hence deduce the existence of a strong solution.

Bounds on the Hausdorff dimension of random attractors for infinite-dimensional random dynamical systems on fractals

<p style='text-indent:20px;'>We consider a stochastic nonlinear evolution equation where the domain is given by a fractal set. The linear part of the equation is given by a Laplacian defined on the fractal. This equation generates a random dynamical system. The long time behavior is given by an attractor which has a finite Hausdorff dimension. We would like to reveal the connections between upper and lower estimates of this Hausdorff dimension and the geometry of the fractal. In particular, the parameter which determines these bounds is the spectral exponent of the fractal. Especially for the lower estimate we construct a local unstable random Lipschitz manifold.